openAI-math-dataset / lean /OAI /RepresentationTheory /UniversalSquare /Support /TransposeSupport.lean
| import Mathlib | |
| import OAI.RepresentationTheory.Saxl.Main | |
| import OAI.RepresentationTheory.UniversalSquare.Support.Candidate | |
| namespace OAI | |
| /-! Transpose Support. -/ | |
| section | |
| noncomputable section | |
| open scoped TensorProduct | |
| namespace UniversalTensorSquare | |
| def signTwistEquiv {n : ℕ} {X Y : Type*} [AddCommGroup X] [Module ℂ X] | |
| [AddCommGroup Y] [Module ℂ Y] | |
| {ρ : Representation ℂ (Equiv.Perm (Fin n)) X} | |
| {σ : Representation ℂ (Equiv.Perm (Fin n)) Y} (e : ρ.Equiv σ) : | |
| (Saxl.signTwist ρ).Equiv (Saxl.signTwist σ) where | |
| toLinearEquiv := e.toLinearEquiv | |
| isIntertwining' := (Saxl.signTwistMap e.toIntertwiningMap).isIntertwining' | |
| lemma canonical_sign_transpose {n : ℕ} (μ : YoungDiagram) (hμ : μ.card = n) : | |
| Nonempty ((Saxl.spechtRep (Saxl.canonicalTableau μ hμ)).Equiv | |
| (Saxl.signTwist (Saxl.spechtRep | |
| (Saxl.canonicalTableau μ.transpose ((Saxl.transpose_card μ).trans hμ))))) := by | |
| obtain ⟨e⟩ := Saxl.specht_sign_transpose (Saxl.canonicalTableau μ hμ) | |
| let e' := Saxl.equivOfSubrepEq | |
| (Saxl.spechtSub (Saxl.transposeTableau (Saxl.canonicalTableau μ hμ))) | |
| (Saxl.spechtSub | |
| (Saxl.canonicalTableau μ.transpose ((Saxl.transpose_card μ).trans hμ))) | |
| (Saxl.spechtSub_tableau_independent _ _) | |
| exact ⟨e.trans (signTwistEquiv e')⟩ | |
| def spechtShapeEquiv {n : ℕ} {μ ν : YoungDiagram} (h : μ = ν) | |
| (t : Saxl.Tableau n μ) (s : Saxl.Tableau n ν) : | |
| (Saxl.spechtRep t).Equiv (Saxl.spechtRep s) := by | |
| subst ν | |
| exact Saxl.equivOfSubrepEq _ _ (Saxl.spechtSub_tableau_independent t s) | |
| lemma canonical_self_sign {n : ℕ} (lam : YoungDiagram) (hlam : lam.card = n) | |
| (hself : lam.transpose = lam) : | |
| Nonempty ((Saxl.spechtRep (Saxl.canonicalTableau lam hlam)).Equiv | |
| (Saxl.signTwist (Saxl.spechtRep (Saxl.canonicalTableau lam hlam)))) := by | |
| obtain ⟨e⟩ := canonical_sign_transpose lam hlam | |
| let e' := spechtShapeEquiv hself | |
| (Saxl.canonicalTableau lam.transpose ((Saxl.transpose_card lam).trans hlam)) | |
| (Saxl.canonicalTableau lam hlam) | |
| exact ⟨e.trans (signTwistEquiv e')⟩ | |
| lemma tensorAnti_injective {n : ℕ} {X Y Z : Type*} | |
| [AddCommGroup X] [Module ℂ X] [AddCommGroup Y] [Module ℂ Y] | |
| [AddCommGroup Z] [Module ℂ Z] | |
| (ρ : Representation ℂ (Equiv.Perm (Fin n)) X) | |
| {σ : Representation ℂ (Equiv.Perm (Fin n)) Y} | |
| {τ : Representation ℂ (Equiv.Perm (Fin n)) Z} | |
| (e : σ.Equiv (Saxl.signTwist τ)) : | |
| Function.Injective (Saxl.tensorAnti ρ e.toIntertwiningMap) := | |
| (TensorProduct.congr (LinearEquiv.refl ℂ X) e.toLinearEquiv).injective | |
| lemma kronecker_pos_transpose {n : ℕ} (lam : YoungDiagram) (hlam : lam.card = n) | |
| (hself : lam.transpose = lam) (ν : YoungDiagram) (hν : ν.card = n) | |
| (hpos : 0 < Saxl.kronecker (Saxl.canonicalTableau lam hlam) | |
| (Saxl.canonicalTableau lam hlam) (Saxl.canonicalTableau ν hν)) : | |
| 0 < Saxl.kronecker (Saxl.canonicalTableau lam hlam) | |
| (Saxl.canonicalTableau lam hlam) | |
| (Saxl.canonicalTableau ν.transpose ((Saxl.transpose_card ν).trans hν)) := by | |
| obtain ⟨eL⟩ := canonical_self_sign lam hlam hself | |
| obtain ⟨eν⟩ := canonical_sign_transpose ν hν | |
| obtain ⟨f, hf⟩ := (Saxl.kronecker_pos_iff _ _ _).mp hpos | |
| let A := Saxl.tensorAnti (Saxl.spechtRep (Saxl.canonicalTableau lam hlam)) | |
| eL.toIntertwiningMap | |
| let B := (Saxl.signTwistMap | |
| (Y := Saxl.Specht (Saxl.canonicalTableau lam hlam) ⊗[ℂ] | |
| Saxl.Specht (Saxl.canonicalTableau lam hlam)) f).comp (Saxl.signCancelMap eν.symm.toIntertwiningMap) | |
| apply (Saxl.kronecker_pos_iff _ _ _).mpr | |
| refine ⟨A.comp B, ?_⟩ | |
| intro hz | |
| apply hf | |
| ext x | |
| have h := congrArg (fun F => F (eν x)) hz | |
| have hi : Function.Injective A := tensorAnti_injective _ eL | |
| apply hi | |
| change A (f x) = A 0 | |
| change A (f (eν.symm (eν x))) = 0 at h | |
| simpa only [eν.symm_apply_apply, map_zero] using h | |
| theorem kronecker_pos_transpose_iff {n : ℕ} (lam : YoungDiagram) (hlam : lam.card = n) | |
| (hself : lam.transpose = lam) (ν : YoungDiagram) (hν : ν.card = n) : | |
| (0 < Saxl.kronecker (Saxl.canonicalTableau lam hlam) | |
| (Saxl.canonicalTableau lam hlam) (Saxl.canonicalTableau ν hν)) ↔ | |
| (0 < Saxl.kronecker (Saxl.canonicalTableau lam hlam) | |
| (Saxl.canonicalTableau lam hlam) | |
| (Saxl.canonicalTableau ν.transpose ((Saxl.transpose_card ν).trans hν))) := by | |
| constructor | |
| · exact kronecker_pos_transpose lam hlam hself ν hν | |
| · intro h | |
| have ht := kronecker_pos_transpose lam hlam hself ν.transpose | |
| ((Saxl.transpose_card ν).trans hν) h | |
| obtain ⟨f, hf⟩ := (Saxl.kronecker_pos_iff _ _ _).mp ht | |
| let e := spechtShapeEquiv (YoungDiagram.transpose_transpose ν).symm | |
| (Saxl.canonicalTableau ν hν) | |
| (Saxl.canonicalTableau ν.transpose.transpose | |
| ((Saxl.transpose_card ν.transpose).trans ((Saxl.transpose_card ν).trans hν))) | |
| apply (Saxl.kronecker_pos_iff _ _ _).mpr | |
| refine ⟨f.comp e.toIntertwiningMap, ?_⟩ | |
| intro hz | |
| apply hf | |
| ext x | |
| have hh := congrArg (fun F => F (e.symm x)) hz | |
| change f (e (e.symm x)) = 0 at hh | |
| simpa using hh | |
| lemma fixedCandidate_support_transpose (n : ℕ) (hM : 3 ≤ staircaseIndex n) | |
| (ν : YoungDiagram) (hν : ν.card = n) : | |
| (0 < Saxl.kronecker (Saxl.canonicalTableau (fixedCandidate n) (fixedCandidate_card n hM)) | |
| (Saxl.canonicalTableau (fixedCandidate n) (fixedCandidate_card n hM)) | |
| (Saxl.canonicalTableau ν hν)) ↔ | |
| (0 < Saxl.kronecker (Saxl.canonicalTableau (fixedCandidate n) (fixedCandidate_card n hM)) | |
| (Saxl.canonicalTableau (fixedCandidate n) (fixedCandidate_card n hM)) | |
| (Saxl.canonicalTableau ν.transpose ((Saxl.transpose_card ν).trans hν))) := | |
| kronecker_pos_transpose_iff _ _ (fixedCandidate_transpose n) ν hν | |
| end UniversalTensorSquare | |
| end | |
| end | |
| end OAI | |